Given that l + m + n + o = 0, which of the given statements are true:

(a) l, m, n and o each must be a null vector.

(b) The magnitude of (l + n) equals the magnitude of (m+ o).

(c) The magnitude of l can never be greater than the sum of the magnitudes of m, n and o.

(d) m + n must lie in the plane of l and o if l and o are not collinear, and in the line of l and o, if they are collinear?

Asked by Pragya Singh | 1 year ago |  62

##### Solution :-

Explanation:-

In order to make l + m + n + o = 0, it is not necessary to have all the four given vectors to be null vectors.

There are other combinations which can give the sum zero.

Explanation:-

l + m + n + o = 0

l + n = – (m + o)

Taking mode on both the sides,

$$\left | l + n \right | = \left | -\left ( m + o \right ) \right | = \left | m + o \right |$$

Therefore, the magnitude of (l + n) is the same as the magnitude of (m + o).

Explanation:-

l + m + n + o = 0

l = (m + n + o)

Taking mode on both the sides,

$$\left | l \right | = \left | m + n + o \right | \left | l \right | \leq \left | l \right | + \left | m \right | + \left | n \right |$$ .......(i)

Equation (i) shows the magnitude of l is equal to or less than the sum of the magnitudes of m, n and o.

Explanation:-

For,

l + m + n + o = 0

The resultant sum of the three vectors l, (m + n), and o can be zero only if (m + n) lie in a plane containing l and o, assuming that these three vectors are represented by the three sides of a triangle.

If l and o are collinear, then it implies that the vector (m + n) is in the line of l and o. This implication holds only then the vector sum of all the vectors will be zero.

Answered by Abhisek | 1 year ago

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